We introduce a collection of polynomials  associated to each positive integer N, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the conjecture, it does suggest two natural generalizations for which we provide some numerical evidence. As these polynomials  are independently interesting, we further explore their basic properties, giving among other things, asymptotic estimates on the growth of their coefficients.

Goldbach conjecture, polynomials, divisibility, irreducible.